Cantor set

Called Under the Cantor set, also cal discontinuum cantor, Cantor dust or wipe quantity is understood in mathematics, a certain subset of the set of real numbers with special topological, measure theoretic, geometric and set-theoretic properties: it is

• Compact, perfect, totally disconnected (a " discontinuum " ) and nowhere dense;
• A Lebesgue -null set;
• Self-similar and has a non-integer Hausdorff dimension (that is a fractal );
• Equally powerful to the continuum ( the set of all real numbers), so in particular uncountable.

It is named after the mathematician Georg Cantor.

For a definition and more detailed descriptions of this set, see below.

General is also called a certain quantity or topological spaces Cantor sets, if they have a part of these properties. Which of these properties are required, it depends on the math area and often also on the context. A topological space which is homeomorphic to the Cantor set, ie the Cantor space.

Construction

The Cantor set can be constructed by the following iteration:

After iterations intervals covering the whole of the original interval exist. The more intervals contains this amount, the lower the percentage of the original interval. The Cantor set now consists of all points that have survived every wiping. In the limiting case ( average of all th wiping quantities ) is the proportion of the original interval zero, although still exist uncountably many elements. This method of construction is used with the Koch curve.

One can describe the interval, which have a representation as a decimal number in base 3, in which only the digits 0 and 2 occur, the Cantor set as the set of all numbers. In particular, the Cantor set includes more than the edge points of the remote intervals; these boundary points are exactly the numbers in, which can be written with a 0 - cycle or a 2- period, for example

Is the left edge point of the first step interval away. The use of the digit 1 is bypassed by the 2 - period, which is the same number. ( This is only possible for one directly in front of the 0 - period. Elsewhere, though no one can occur because otherwise the number would be in the middle of one of the deleted intervals. ) But In addition, for example, also fourth in the Cantor set, because

Properties

Be the Hausdorff dimension and Minkowski dimension of the Cantor set

0-1- sequences

The Cartesian product of countably infinitely many copies of the two-element set is the set of all infinite sequences that take on the values ​​0 and 1, ie the set of all functions. This quantity is denoted by. By the above triadic development can be a natural bijection between the Cantor set and the quantity: The number associated with the triadic development is translated into the sequence; the number of 1/4 corresponds to the result.

The amount also bears a natural topology (i.e. the product topology induced by the topology of the discrete quantity). The just -mentioned figure is a homeomorphism between the Cantor set and the topological space. This is therefore called the Cantor space.

Cantor and Cantor distribution function

Closely related to the Cantor set is the Cantor distribution. It is constructed similar to the Cantor set. Its distribution function is also known Cantor function.

Cantor distribution is often used as an example of the existence of the singular continuous distributions which are unique with respect to the Lebesgue measure, but have a continuous distribution function ( function with a so-called singular continuous behavior).

Other Cantor sets

The Cantor set (also middle thirds Cantor set, middle Thirds Cantor set) described above. By a Cantor set is defined as a set of real numbers, one gets a variation of the above wiping process, where you can now vary the lengths and numbers of weggewischten intervals:

One begins with an arbitrary interval of real numbers. In the first step removed finitely many open disjoint intervals (but at least one ) and obtains a finite number of closed intervals ( at least 2).

In the second step is removed from each of the intervals contains a finite number of sub-intervals in turn ( at least one).

Again, this process defines a set of real numbers, namely those points that never fell into one of the weggewischten intervals.

One can show that all so constructed Cantor sets are homeomorphic to each other and that they are particularly suitable for the set of all real numbers equally powerful. By the proportion " lengths of the intervals weggewischten: lengths of the remaining intervals" suitably varied, one can generate a Cantor set whose Hausdorff dimension is any given number in the interval [ 0,1].

A two-dimensional analogue of the Cantor set, the Sierpinski carpet, a three-dimensional the Menger sponge.